cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A159315 E.g.f. satisfies: d/dx log(A(x)) = A(2*x)^(1/2).

Original entry on oeis.org

1, 1, 2, 7, 41, 406, 7127, 235147, 15191966, 1953128401, 501361942127, 257110692345262, 263513099974512041, 539923433830720468321, 2212048542930121133510402, 18123271334339868892408048927
Offset: 0

Views

Author

Paul D. Hanna, Apr 19 2009

Keywords

Comments

Row 0 of array A159314.

Examples

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 41*x^4/4! + 406*x^5/5! +...
Related expansions:
log(A(x)) = x +x^2/2! +3*x^3/3! +19*x^4/4! +225*x^5/5! +4801*x^6/6! +...
A(2*x)^(1/2) = 1 + x + 3*x^2/2! +19*x^3/3! +225*x^4/4! +4801*x^5/5! +...
in which the coefficients are given by A126444.

Links

Crossrefs

Cf. A159314, A126444, A159316, A159317.

Programs

  • PARI
    {a(n)=local(A=vector(n+2, j, 1+j*x)); for(i=0, n+1, for(j=0, n, m=n+1-j; A[m]=exp(intformal((A[m+1]+x*O(x^n))^(2^(m-1)))))); n!*polcoeff(A[1], n, x)}

Formula

E.g.f. satisfies: A'(x) = A(x)*A(2*x)^(1/2).
a(n) = Sum_{i=0..n-1} C(n-1,i)*A126444(i)*a(n-1-i) for n>0 with a(0)=1.
E.g.f.: A(x) = G(x/2)^2 where G(x) = e.g.f. of A126444.
E.g.f.: A(x) = F(x/4)^4 where F(x) = e.g.f. of A159316.
a(n) ~ c * 2^(n*(n-3)/2), where c = 14.6416352593041803546... - Vaclav Kotesovec, Feb 23 2014

A159314 Rectangular array, read by antidiagonals, where row e.g.f.s, R(n,x), satisfy: d/dx log( R(n,x) ) = R(n+1,x)^(2^n) with R(n,0) = 1; that is, the logarithmic derivative of the e.g.f. of row n equals the e.g.f. of row n+1 to the 2^n power, for n>=0.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 7, 1, 1, 5, 19, 41, 1, 1, 9, 61, 225, 406, 1, 1, 17, 217, 1481, 4801, 7127, 1, 1, 33, 817, 10737, 66361, 185523, 235147, 1, 1, 65, 3169, 81761, 988561, 5390285, 13298659, 15191966, 1, 1, 129, 12481, 638145, 15269281, 164637369
Offset: 0

Views

Author

Paul D. Hanna, Apr 19 2009

Keywords

Examples

			Array begins:
1,1,2,7,41,406,7127,235147,15191966,1953128401,501361942127,...;
1,1,3,19,225,4801,185523,13298659,1815718305,481790947681,...;
1,1,5,61,1481,66361,5390285,803252341,224927827601,...;
1,1,9,217,10737,988561,164637369,49987302697,28333326990177,...;
1,1,17,817,81761,15269281,5149256177,3155353490257,...;
1,1,33,3169,638145,240072001,162919458273,200565037419169,...;
1,1,65,12481,5042561,3807826561,5184101454785,12792473234253121,...;
1,1,129,49537,40092417,60660860161,165425163421569,...;
1,1,257,197377,319751681,968467745281,5286172203486977,...;
1,1,513,787969,2554072065,15478671283201,169038775947894273,...;
1,1,1025,3148801,20416829441,247524381173761,5407342625815542785,...;
...
where row e.g.f.s begin:
R(0,x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 41*x^4/4! + 406*x^5/5! +...;
R(1,x) = 1 + x + 3*x^2/2! +19*x^3/3! +225*x^4/4! +4801*x^5/5! +...;
R(2,x) = 1 + x + 5*x^2/2! +61*x^3/3!+1481*x^4/4!+66361*x^5/5! +...;
...
Row e.g.f.s satisfy: R(n+1,x)^(2^n) = d/dx log( R(n,x) ):
R(1,x)^1 = d/dx log(1+x +2*x^2/2! +7*x^3/3! +41*x^4/4! +...);
R(2,x)^2 = d/dx log(1+x +3*x^2/2! +19*x^3/3! +225*x^4/4! +...);
R(3,x)^4 = d/dx log(1+x +5*x^2/2! +61*x^3/3! +1481*x^4/4! +...);
R(4,x)^8 = d/dx log(1+x +9*x^2/2! +217*x^3/3! +10737*x^4/4! +...);
...
Examples of R(n,x) = R(n+m,x/2^m)^(2^m):
R(n-1,x) = R(n,x/2)^2 and R(n+1,x) = R(n,2x)^(1/2);
R(0,x) = R(n,x/2^n)^(2^n) and R(n,x) = R(0,2^n*x)^(1/2^n).

Crossrefs

Cf. rows: A159315, A126444, A159316, diagonal: A159317, variant: A145085.

Programs

  • PARI
    {T(n,k)=if(k==0,1,sum(i=0,k-1,2^(n*i)*binomial(k-1,i)*T(1,i)*T(n,k-1-i)))}
  • PARI
    {T(n, k)=local(A=vector(n+k+2, j, 1+j*x)); for(i=0, n+k+1, for(j=0, n+k, m=n+k+1-j; A[m]=exp(intformal((A[m+1]+x*O(x^k))^(2^(m-1)))))); k!*polcoeff(A[n+1], k, x)}

Formula

T(n,k) = Sum_{i=0..k-1} C(k-1,i)*2^(n*i)*T(1,i)*T(n,k-1-i) for k>0 with T(n,0)=1, for n>=0.
Row e.g.f.s, R(n,x), satisfy:
(1) R'(n,x)/R(n,x) = R(n+1,x)^(2^n) with R(n,0) = 1;
(2) R(n,x) = R(n+m,x/2^m)^(2^m) for m >= -n.
Showing 1-2 of 2 results.

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